$ F = \left[\begin{array}{rrr}-2 & -2 & -1 \\ 1 & 2 & 1\end{array}\right]$ $ v = \left[\begin{array}{r}-2 \\ 3 \\ 5\end{array}\right]$ What is $ F v$ ?
Solution: Because $ F$ has dimensions $(2\times3)$ and $ v$ has dimensions $(3\times1)$ , the answer matrix will have dimensions $(2\times1)$ $ F v = \left[\begin{array}{rrr}{-2} & {-2} & {-1} \\ {1} & {2} & {1}\end{array}\right] \left[\begin{array}{r}{-2} \\ {3} \\ {5}\end{array}\right] = \left[\begin{array}{r}? \\ ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ F$ , with the corresponding elements in column $j$ of the second matrix, $ v$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ F$ with the first element in ${\text{column }1}$ of $ v$ , then multiply the second element in ${\text{row }1}$ of $ F$ with the second element in ${\text{column }1}$ of $ v$ , and so on. Add the products together. $ \left[\begin{array}{r}{-2}\cdot{-2}+{-2}\cdot{3}+{-1}\cdot{5} \\ ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ F$ with the corresponding elements in ${\text{column }1}$ of $ v$ and add the products together. $ \left[\begin{array}{r}{-2}\cdot{-2}+{-2}\cdot{3}+{-1}\cdot{5} \\ {1}\cdot{-2}+{2}\cdot{3}+{1}\cdot{5}\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{r}{-2}\cdot{-2}+{-2}\cdot{3}+{-1}\cdot{5} \\ {1}\cdot{-2}+{2}\cdot{3}+{1}\cdot{5}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{r}-7 \\ 9\end{array}\right] $